This page features exercises focused on \(LU\) factorizations of various matrices, showcasing the decomposition process into lower and upper triangular forms. It includes solving systems of linear equ...This page features exercises focused on \(LU\) factorizations of various matrices, showcasing the decomposition process into lower and upper triangular forms. It includes solving systems of linear equations through \(LU\) factorization, detailing intermediate steps and solutions.
Note that \(x^{T}Ax\) is always a scalar value (e.g., note that \(x^TA = y^T\) for some vector \(y\in\mathbb{R}^n\), and \(y^Tx\) is the dot product between \(x\) and \(y\) and, hence, a real scalar)....Note that \(x^{T}Ax\) is always a scalar value (e.g., note that \(x^TA = y^T\) for some vector \(y\in\mathbb{R}^n\), and \(y^Tx\) is the dot product between \(x\) and \(y\) and, hence, a real scalar). Exists for every square matrix and says every such matrix, \(A\), is unitarily equivalent to an upper-triangular matrix, \(T\) (i.e., there exists an orthonomal basis with respect to which \(A\) is upper-triangular)
